Related papers: Topology-Preserving Scaling in Data Augmentation
We numerically reexamine the scaling behavior of period doublings in four-dimensional volume-preserving maps in order to resolve a discrepancy between numerical results on scaling of the coupling parameter and the approximate…
We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window (or time-delay) embeddings, as seen through the lens of…
Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation…
Model merging has emerged as a promising paradigm for enabling multi-task capabilities without additional training. However, existing methods often experience substantial performance degradation compared with individually fine-tuned models,…
Topology and machine learning are two actively researched topics not only in condensed matter physics, but also in data science. Here, we propose the use of topological data analysis in unsupervised learning of the topological phase…
Geometric data augmentation is widely used in segmentation workflows, but polygon annotations are often assumed to remain valid after transformation. This assumption can fail in structured domains such as architectural floorplan analysis,…
The scaling argument developed by Larichev and Held (1995) for eddy amplitudes and fluxes in a horizontally homogeneous, two-layer model on an f-plane is extended to a beta-plane. In terms of the non-dimensional number x =…
In this paper, we introduce the persistence transformation, a novel methodology in Topological Data Analysis (TDA) for applications in time series data which can be obtained in various areas such as science, politics, economy, healthcare,…
Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially…
Dimensionality reduction techniques are widely used for visualizing high-dimensional data in two dimensions. Existing methods are typically designed to preserve either local (e.g., $t$-SNE, UMAP) or global (e.g., MDS, PCA) structure of the…
One of the difficulties of training deep neural networks is caused by improper scaling between layers. Scaling issues introduce exploding / gradient problems, and have typically been addressed by careful scale-preserving initialization. We…
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…
In this paper, we propose a unified algorithmic framework for solving many known variants of \mds. Our algorithm is a simple iterative scheme with guaranteed convergence, and is \emph{modular}; by changing the internals of a single…
Modern business and economic datasets often exhibit nonlinear, multi-scale structures that traditional linear tools under-represent. Topological Data Analysis (TDA) offers a geometric lens for uncovering robust patterns, such as connected…
We present an alternative finite-size approach to a set of parity conserving interfaces involving attachment, dissociation, and detachment of extended objects in 1+1 dimensions. With the aid of a nonlocal construct introduced by Barma and…
We give two low-complexity algorithms, one for dimensionality reduction and one for dimensionality increase, which are applicable to any dataset, regardless of whether the set has an intrinsic dimension or not. The corresponding methods…
Optimization, a key tool in machine learning and statistics, relies on regularization to reduce overfitting. Traditional regularization methods control a norm of the solution to ensure its smoothness. Recently, topological methods have…
In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a…